Credit Rating Dynamics and Competition

Transcription

1 Working Paper, February 20, 2013, Pages 1 33 Credit Rating Dynamics and Competition Stefan Hirth Aarhus University, Business and Social Sciences, Fuglesangs Allé 4, DK 8210 Aarhus V, Denmark. shirth@econ.au.dk, phone: Abstract. I analyze credit rating agencies and competition on a market with more than two agencies. Both investors and agencies react to each other s behavior. My model predicts cyclic dynamics in the base case: not only does the presence of trusting investors facilitate ratings inflation. In turn, ratings inflation also induces investors to be less trusting. A regulator can implement honest rating behavior by abolishing the issuer pays model or by a centralized monitoring of ratings quality. The effect of the entry or exit of an agency on ratings quality depends on the current number of rating agencies on the market. JEL Classification: D43, D82, G24, L15 Keywords: credit rating agencies, ratings inflation, evolutionary game theory. A previous version of this paper was entitled Beyond Duopoly: The Credit Ratings Game Revisited. I thank David M. Arseneau, Andreas Barth, Patrick Bolton, Matthias Jüttner, Anastasia V. Kartasheva, David Lando, Martin Oehmke, Marcus M. Opp, Henri Pagès, Martin Ruckes, Joel Shapiro, Florina Silaghi, Konrad Stahl, Günter Strobl, Tim Thabe, Olivier Toutain, Marliese Uhrig-Homburg, and Martin Zenker for fruitful comments and discussions. Moreover, I thank conference participants at Danish Convent on Accounting and Finance, Copenhagen, Denmark 2012, German Finance Association, Hannover, Germany 2012, French Finance Association, Strasbourg, France 2012, Financial Risks International Forum on Systemic Risk, Paris, France 2012, Symposium on Finance, Banking, and Insurance, Karlsruhe, Germany 2011, Globalization: Strategies and Effects, Kolding, Denmark 2011, 10th International Conference on Credit Risk Evaluation, Venice, Italy 2011, CEPR European Summer Symposium on Financial Markets, Gerzensee, Switzerland 2011, and FRG Aarhus Research Workshop, Skagen, Denmark 2010, as well as seminar participants at Europlace Institute of Finance, Paris, France 2013, Universität Münster, Germany 2013, Universität Ulm, Germany 2012, Vrije Universiteit Amsterdam, Netherlands 2012, University of St. Gallen, Switzerland 2012, Banque de France, Paris, France 2012, Johannes Gutenberg-Universität, Mainz, Germany 2012 and KIT Workshop on Economics and Finance, Karlsruhe, Germany 2011 for their valuable feedback on my work, and Annette Mortensen for proofreading. I gratefully acknowledge financial support from the Europlace Institute of Finance. Part of this work has been conducted while I was a visiting scholar at Columbia University in spring I thank Suresh Sundaresan for inviting me, and I gratefully acknowledge financial support for this visit from Aarhus University and Otto Mønsteds Fond.

2 2 Stefan Hirth 1. Introduction How does the complex interaction between credit rating agencies (CRAs), issuers, and investors affect the quality and informativeness of credit ratings? CRAs are widely considered to have been a major factor within the development of the 2008 subprime mortgage crisis, accused of intentionally inflating ratings, i.e., giving good ratings to bad issues. Most recently, the U.S. Justice Department charged the largest CRA, Standard & Poor s (S&P), with fraud and demanded US$5-billion in restitution in early February 2013, see Mattingly (2013). In this paper, I aim to answer the question how honest rating behavior can be achieved. First, I analyze how investor behavior affects the behavior of CRAs and vice versa. Second, I make predictions about how this interaction is affected by the competitiveness of the market for credit ratings, as measured by the number of CRAs on the market. Third, I provide implications for policy makers how regulation can support honest rating behavior. The U.S. market is characterized by a limited number of approved CRAs, so-called Nationally Recognized Statistical Rating Organizations (NRSRO). First there were only Moody s and S&P, and since approximately 1997, Fitch has been there as the third agency. In the meantime, seven more agencies have been approved, so there are now ten CRAs that are designated as NRSROs. 1 ThethreebigCRAshavemorethan90percentof the market share, see Atkins (2008). In earlier years, the NRSRO designation possibly was a barrier of entry. However, the current situation with ten NRSROs on the market and seven of them not significantly improving their market shares suggests that further analysis is needed: within a theoretical framework that allows for ten or more agencies, it remains to be answered under which conditions players with negligible market shares can improve their positions. 2 It might be particularly interesting to determine the conditions under which a new rating agency that possibly has different ethical standards and business practices can successfully invade the market, even if it starts off with a tiny market share. The current market for credit ratings is dominated by the issuer pays business model. 3 It was switched from an earlier investor pays model due to information drain and difficulties in collecting sufficient fees. 4 However, the issuer pays model has an inherent 1 See retrieved February 2, Jeon and Lovo (2011) suggest that apart from the approval as NRSRO, natural barriers of entry can be a critical factor and prevent new CRAs from being successful against the incumbent. 3 In the present paper, I consider only the case of solicited credit ratings, i.e., that the issuer pays for receiving a rating. See, e.g., Bannier, Behr, and Güttler (2010) and Fulghieri, Strobl, and Xia (2012) for a discussion of unsolicited credit ratings, i.e., those provided by CRAs without receiving compensation from the market. For the U.S. market, Gan (2004) estimates that unsolicited ratings account for 22% of all new issue ratings between 1994 and There are some smaller CRAs on the market that apply the investor pays model. One of them, namely Egan-Jones Ratings Company, is even an NRSRO. A recent investigation by the U.S. Securities and Exchange Commission (SEC) against Egan-Jones shows that the investor pays model is not necessarily free of conflicts of interest either, see SEC (2012). The theoretical model by Stahl and Strausz (2011) suggests that the issuer pays business model might be superior to the investor pays model. They argue in a more general context that sellers (issuers) rather than buyers (investors) of an information-sensitive

3 Credit Rating Dynamics and Competition 3 conflict of interest: It can be profitable for the CRA to inflate ratings. 5 A related problem is ratings shopping: The issuer chooses to pay the fee only to a CRA that promises to give a favorable rating, and approaches a competitor otherwise. The question whether more competition can increase ratings quality is thus related to whether it helps to prevent ratings inflation, although it can give more opportunities for ratings shopping. As two shortcomings of existing theoretical models, I identify that they only consider competition in duopoly and usually neglect the dynamic properties of the market. This gap that my paper aims to fill is well motivated by quoting Bolton, Freixas, and Shapiro (2012) (hereafter BFS), p.104: It would be of interest (but beyond the scope of [their] paper) to explore these issues more systematically in a fully general dynamic game, possibly with an infinite horizon. There is currently no model of oligopolistic competition over an infinite horizon in the CRA literature; indeed, there are very few such models in the industrial organization literature for obvious reasons of tractability. Therefore I develop a tractable framework using Evolutionary Game Theory to analyze the interaction of CRAs over an infinite horizon in a competitive market with an arbitrary number of agencies. I model the CRAs incentives to inform the investors honestly about the quality of investments, rather than to inflate ratings, as an interplay with investors sophistication level. These characteristics of CRAs and investors are similarly modeled in BFS and other papers. As the main innovation on the modeling side, the methodology of Evolutionary Game Theory allows for an arbitrary number of market participants, as well as the analysis whether new behavioral traits successfully enter an established market. As an important contribution, I do not only consider changes in CRA behavior, but I also allow the behavior of the investors to react on the characteristics of the CRA market they face. On the one hand, CRAs are more likely to inflate ratings for a high share of trusting investors, as the benefits from receiving more fees outweigh the possible reputation costs if caught inflating. On the other hand, investors will in turn change their behavior when they face a CRA market with a lot of ratings inflation. As sophisticated investors perform better than trusting investors on such a market, the latter will either start leaving the market or learn to be more sophisticated as well. In my model, sophisticated investors have to spend costs for the monitoring of investment and ratings quality, whereas trusting investors save these costs but suffer when they happen to buy bad investments. First, I show that the interaction between the CRAs and the investors behavior can lead to different equilibria. Dependent on the parametrization, either honest or inflating CRAs dominate in the end, while the population of investors ends up being either trusting or sophisticated. There can even be cyclic dynamics with the distribution shares in both populations periodically increasing and decreasing over time. good should pay for certification (ratings). While sellers want to signal quality, buyers have to inspect quality, the former being both socially more desirable and generating higher rents to the certifier (CRA). 5 Throughout my paper I assume that CRAs have perfect knowledge about the actual quality of investments, and the only remaining issue is whether they truthfully report information to the investors. This assumption is questionable, especially in the light of recent underperformance when rating structured products. Among others, Pagano and Volpin (2010) investigate the interplay of ratings inflation and the failure of CRAs to provide accurate ratings, and Bar-Isaac and Shapiro (2012) discuss how ratings quality is related to analyst skills. Pagano and Volpin (2012) show that issuers may prefer (and CRAs produce) coarse and uninformative ratings even in the absence of ratings inflation and ratings shopping.

4 4 Stefan Hirth In a second step, I make additional assumptions about how the model s parameters could depend on the number of CRAs in the market. Thus, I can show how different market structures and outcomes result from changing the number of CRAs. Existing theoretical models can only distinguish between a monopolistic CRA and competition in duopoly. In contrast, I show that increasing competition can lead to significant changes in market structures and outcomes for any arbitrary current market size, for example when one new CRA enters a market currently consisting of two, three, or ten CRAs. I postulate that for example reputation costs increase with the number of CRAs on the market, as the investors threat of punishing inflating CRAs becomes more effective. In this case there is a critical number of CRAs on the market, above which the reputation costs are high enough such that honest rating behavior pays off at least temporarily. Finally, my analysis leads to the following two policy recommendations, without explicitly addressing the number of CRAs on the market: First, it is essential to find an alternative solution to the issuer pays model, and particularly to prevent that rating agencies can achieve higher revenues by issuing good ratings. Second, the monitoring of CRAs performance and their possible punishment should rather be done (even more so) by a regulator or a central market authority, rather than individual investors. If at least one of these issues can be solved, then the market for credit ratings will function well in the sense that honest rating behavior is viable, independent of the size of the CRA market. Regarding the optimal number of CRAs on an oligopolistic market, I show that the entry or exit of an agency can change the equilibrium outcome and has an ambiguous effect on ratings quality. The direction of the effect depends on the current number of CRAs on the market. Most researchers agree that ratings inflation is most severe for complex investment products as in structured finance, see e.g. Skreta and Veldkamp (2009) and BFS. On the other hand, Baghai, Servaes, and Tamayo (2013) show that CRAs have even become more conservative in assigning corporate credit ratings. In line with this, my paper should also be understood in the context of rating complex investment products rather than corporations. 6 Mathis, McAndrews, and Rochet (2009) analyze ratings inflation for a monopolistic CRA. They derive a reputation cycle similar to the cyclic dynamics in the base case of my model, however they exclude the possibility of the CRA to recover its reputation, once it is caught lying. In general, the existing literature comes to ambiguous conclusions on the relation between competition and ratings inflation. 7 Camanho, Deb, and Liu (2012) extend Mathis, McAndrews, and Rochet (2009) by competition effects. They find that competition results in greater ratings inflation. Similarly, Skreta and Veldkamp (2009) and BFS find that competition makes ratings shopping worse. On the contrary, Manso (2012) highlights that credit ratings can have feedback effects on the credit quality of issuers. He finds that increased competition between rating agencies can create downward pressure on ratings and tougher rating policies. Doherty, Kartasheva, and Phillips (2012) show both theoretically and empirically that the market entry of a new CRA can improve ratings quality and precision. Their story is that the entrant CRA can attract business from good issuers that have been pooled with worse quality issuers. By using a more 6 I thank David Lando for pointing this out. 7 Apart from competition effects, ratings inflation is influenced by other factors. For example, Opp, Opp, and Harris (2012) explain how rating-contingent regulation can contribute to ratings inflation. Stolper (2009) suggests that the problem might be solved by a proper regulatory approval scheme for CRAs.

5 Credit Rating Dynamics and Competition 5 precise rating scale, the entrant CRA allows the good issuers to receive higher prices for the investments they sell. Becker and Milbourn (2011) take the market entry of Fitch as a natural experiment to analyze the effect of increasing competition. Overall, they show a decrease in ratings quality. Bongaerts, Cremers, and Goetzmann (2012) suggest a different role of Fitch, namely being a tiebreaker if the other two big rating agencies, S&P and Moody s, disagree whether a bond issue has investment grade or high yield status. Assuming that a Fitch rating is solicited more often if the issuer expects it to break the tie towards investment grade, this endogeneity provides an alternative to the ratings inflation story, explaining why the observed Fitch ratings are higher on average. Similarly, Xia (2012) provides evidence in the opposite direction of Becker and Milbourn (2011). He shows that increased competition by the entry of investor-paid Egan-Jones Rating Company led to higher ratings quality for S&P. He, Qian, and Strahan (2011) examine whether rating agencies reward large issuers of mortgage-backed securities. After controlling for deal characteristics, they can analyze a situation in which small and large issuers differ only in the amount of possible future business. They find evidence for a positive bias of CRAs towards large issuers and thus for ratings inflation. Similarly, Mählmann (2011) shows that firms with longer rating agency relationships have better credit ratings, although they do not have lower default rates. Hörner (2002) provides a general reputational theory in which competition can increase quality if the consumers competitive choice makes loss of reputation a real threat. One of the first papers analyzing the trade-off between building up a long-term reputation and making higher short-term profits by misbehaving is by Klein and Leffler (1981). So in principle, if more competition should be a cure of ratings inflation, it would need to affect this trade-off towards the benefit of building up a long-term reputation. Related to Klein and Leffler (1981) and Hörner (2002), my hypothesis is that reputation costs are too low in a market with a small number of CRAs. The investors and issuers need to have a sufficient number of alternatives. Then the loss of reputation is a real threat that can induce honest rating behavior. The transition from monopoly to duopoly, or even to a market with three participants, still may not provide sufficient alternatives. This can explain that Becker and Milbourn (2011) do not find an increase in ratings quality following the market entry of Fitch. However, the change could come for an even larger number of CRAs, some of whom possibly offering true alternatives to the existing ones. The paper is structured as follows: I introduce the modeling framework in Section 2.. In Section 3., I visualize and discuss the results for an arbitrary number of CRAs. Next, I discuss the effect of competition and explicitly focus on the number of CRAs in Section 4.. Then I provide empirical implications in Section 5.. In Section 6., I critically investigate limitations and present extensions of my model. Section 7. concludes the paper. 2. Model The model is based on the methodology of Evolutionary Game Theory. More precisely, I build on the replicator dynamics as given in Taylor and Jonker (1978) and Taylor (1979), as well as Schuster, Sigmund, Hofbauer, and Wolff (1981), who derive additional results for evolutionary games between two populations. The economic setting of the model is given for the duopolistic case by BFS, from which I also use the notation as far as possible to ensure comparability.

6 6 Stefan Hirth 2.1 SETUP I consider a market on which issuers provide two types of investments. First, a good investment, which is present on the market with a share λ [0, 1]. It yields a payoff 1+R>1 upon investment of 1, i.e., a net payoff of R>0. Second, a bad investment, which is present on the market with a share (1 λ). It yields a payoff of zero, which can be interpreted as default, upon investment of 1. So the net payoff of the bad investment is ( 1). Apart from the issuers, there are two populations that interact with each other. First, I consider the population of investors (Inv). There is a share α [0, 1] of trusting investors (T), and a share (1 α) of sophisticated investors (S). Second, I consider the population of rating agencies (CRAs). There is a share β [0, 1] of honest CRAs (H), and a share (1 β) of inflating CRAs (I). The population space thus consists of all possible states (α, β) within the square (0,0), (1,0), (1,1), (0,1). I concentrate on the interplay between investors and rating agencies, while taking the remainder of the market as exogenous. As discussed below, the issuers ratings shopping behavior is also captured by my model, although the issuers are not modeled as active players. Also, I provide suggestions that a regulator should follow to change the boundary conditions of the market. 2.2 INTERACTIONS For each interaction, there is a random investment from a given issuer to be rated, which is good with probability λ. Then the interaction takes place in a random pairing of one investor (type T or S) and one CRA (type H or I). The players cannot recognize each other s types. Depending on the players types, they receive payoffs as derived in the following. The CRA charges a fee Φ 0 from the issuer of the investment for giving the rating. 8 The fee is received only for a good rating. This assumption is common in the literature and can be interpreted as a reduced-form modeling of ratings shopping, as the issuer will then move on and hope to find another agency who promises to give him the good rating. Therefore one could argue that the issuer s behavior, namely the choice to accept and pay only for good ratings, is modeled here in a simple form. Effectively, an investment without rating is equivalent to an investment rated as bad in my model. I assume that CRAs can perfectly observe whether investments are good or bad. An honest CRA truthfully reports the type of the investment. Thus, if the investment is bad, it cannot sell the rating to the issuer and does not receive a fee. An inflating CRA, in contrast, always reports good and receives the fee. 8 First, the fee is assumed exogenous and independent of the market conditions. In Section 4., I will allow the fee (and other parameters) to be a function of the number of CRAs on the market. In the Appendix, Section A1, I analyze the case in which the fees that issuers are willing to pay to the CRAs for a good rating depend on the fraction of trusting investors on the market. In my model, there are no costs for the CRAs to produce ratings. However, Φ can also be interpreted as the net fee, i.e., what is remaining for the CRA after production costs, given that all CRAs face the same production costs. I discuss in the Appendix, Section A2 the case of different production costs for inflating and honest CRAs.

7 Credit Rating Dynamics and Competition 7 On the investor side, trusting investors cannot judge the investments. They buy all investments that are rated as good. In contrast, sophisticated investors spend a cost C 0 to verify the CRA s work and evaluate the investments. If they meet an inflating CRA with a bad investment rated as good, they do not buy it, and at the same time they cause reputation costs ρ 0 for the CRA. Here, I make a strong assumption that lying CRAs can immediately be recognized and punished. A more realistic assumption would be that suchbehaviorcanonlybedetectedwithsomedelay,ifatall.however,theassumption is consistent with the previous assumption that investments turn out to be good or bad (without any uncertainty due to overlapping realizations of the outcomes) immediately after making the investment decision. A related critical assumption is that the sophisticated investors still spend monitoring costs to observe the CRAs behavior, although they can perfectly verify the quality of the investments themselves. It can be motivated by assuming that only by the combination of the information they receive from the CRA and their own verification efforts, the sophisticated investors are able to make such a perfect judgment of the investments. 2.3 PAYOFFS There are several possible economic explanations for the population shares changing over time, dependent on the realized payoffs. As a result of the realized payoffs, individuals switch their behavior towards another pure strategy, unsuccessful market participants leave the market, or new market entrants observe and imitate the most successful behavior Investors Regarding the payoffs for the investors, I first consider the trusting investors. If they are meeting an honest CRA, they receive a good rating for a good investment, which occurs with probability λ. In this case they invest and receive a net payoff of R. If the investment is bad, the investors are warned, as the honest CRA refuses to give a good rating, so they do not invest and receive zero payoff. Together, the expected payoff is V TH = λr. (1) Against an inflating CRA, they receive a net payoff of R for a good investment, which occurs with probability λ. However, if the investment is bad, which occurs with probability (1 λ), the CRA still gives a good rating. The trusting investors invest, and consequently receive a net payoff of ( 1). Together, their expected payoff is The resulting expected payoff for trusting investors is V TI = λr +(1 λ)( 1). (2) Π Inv T = βv TH +(1 β)v TI = λr (1 β)(1 λ), (3) as they meet an honest (inflating) CRA with probabilities β and 1 β, respectively. Second, consider the sophisticated investors. If they are meeting an honest or inflating CRA,

8 8 Stefan Hirth Table I Investors Payoffs. The investors payoffs are summarized for each possible investor/cra strategy pair, the investors being either trusting or sophisticated and the CRA being either honest or inflating. In addition, the last column shows the expected payoffs for either type of investor. Investor / CRA honest inflating expected trusting V TH = λr V TI = λr +(1 λ)( 1) Π Inv T = λr (1 β)(1 λ) sophisticated V SH = λr C V SI = λr C Π Inv S = λr C they receive the same payoff, 9 namely V SH = V SI = λr C. (4) In either case, they spend the cost C to verify the CRA s work. Thus, they manage to invest only in the good investments, which occur with probability λ. The resulting expected payoff for sophisticated investors is the same, namely Π Inv S = βv SH +(1 β)v SI = λr C. (5) The resulting payoffs are summarized in Table I. The average payoff in the population of investors is Π Inv = απ Inv T +(1 α)π Inv S. (6) Rating Agencies Considering the rating agencies, I first state the payoffs for the honest CRAs. They give a good rating and receive the fee only if they observe a good investment, which occurs with probability λ. 10 On the other hand, they are never punished for inflating ratings. Their expected payoff against both trusting and sophisticated investors is therefore X HT = X HS = λφ. (7) Thus, the resulting expected payoff for honest CRAs is the same, namely Π CRA H = αx HT +(1 α)x HS = λφ. (8) Second, consider the inflating CRAs. If they are meeting a trusting investor, they receive X IT =Φ. (9) As they always give good ratings, they are always paid by the issuers, regardless of the quality of the investment. Against a sophisticated investor, however, the inflating CRAs 9 As an alternative assumption, one can justify that V SH = λ(r C), while still V SI = λr C. The rationale is that within the model, an observed bad rating indicates an honest CRA, and therefore no monitoring costs have to be spent in that case. I thank Henri Pagès for suggesting this alternative. A motivation for my original modeling can be that there are more than two rating classes in practice, and then it cannot be inferred from observing a specific rating (unless it is the lowest one) whether the CRA is honest. 10 As an alternative specification, one could assume that honest CRAs spend some production costs for rating the investment, while inflating CRAs do not. See the Appendix, Section A2 for a discussion of this case.

9 Credit Rating Dynamics and Competition 9 expected payoff is Table II CRAs Payoffs. The CRAs payoffs are summarized for each possible investor/cra strategy pair, the investors being either trusting or sophisticated and the CRA being either honest or inflating. In addition, the last row shows the expected payoffs foreithertypeofcra. Investor / CRA honest inflating trusting X HT = λφ X IT =Φ sophisticated X HS = λφ X IS =Φ (1 λ)ρ expected Π CRA H = λφ Π CRA I =Φ (1 α)(1 λ)ρ X IS =Φ (1 λ)ρ. (10) While the issuer still pays them the fee regardless of the quality of the investment, they are punished whenever they rate a bad investment as good, which happens with probability (1 λ), and meet a sophisticated investor. The resulting expected payoff for inflating CRAs is Π CRA I = αx IT +(1 α)x IS =Φ (1 α)(1 λ)ρ. (11) The payoffs are summarized in Table II. The average payoff in the population of CRAs is Π CRA = βπ CRA H +(1 β)π CRA I. (12) 2.4 TWO-PLAYER GAME BETWEEN INVESTOR AND CRA Before analyzing the evolutionary game between the two populations of investors and CRAs, I present the related two-player game between one investor and one CRA, with the payoffs given in Tables I and II. I show the game in the normal form, which is the combination of the two payoff matrices, in Table III. As usually done in two-player game theory, the first and second entry represent the payoffs for the investor (column player) and the CRA (row player), respectively. Throughout the evolutionary dynamics, I will assume that for each interaction, there is a random draw of one investor and one CRA with built-in types out of their respective populations. In contrast, for the current section I assume that both the investor and the CRA each choose their optimal strategies. They can either choose a pure strategy, i.e., perform one of their two respective actions with certainty, or choose a mixed strategy, which is to randomize their actions. In the latter case, the investor chooses to be trusting with probability α, and the CRA chooses to be honest with probability β, respectively. The limit cases of choosing probabilities 0 or 1 reflect the pure strategies. As standard in Game Theory, I check for the existence of Nash Equilibria, i.e., strategies that are the best responses to each other s choices. I restrict the attention to cases in which there are at least some bad investments on the market (λ <1), and being caught lying causes positive reputation costs for the CRA (ρ >0). Otherwise, there is either no role for rating investments, or no advantage of being an honest CRA in the model. Table IV summarizes the resulting possible cases. For the interior case with 0 <C<1 λ and 0 < Φ <ρ, there are no Nash equilibria in pure strategies. However, there is a Nash equilibrium in mixed strategies with the

10 10 Stefan Hirth Table III Two-Player Game in Normal Form. For each possible investor/cra strategy pair, the investors being either trusting or sophisticated and the CRA being either honest or inflating, the table states the payoffs for the investor and CRA, respectively, as a pair separated by commas in each cell. Investor / CRA honest inflating trusting λr, λφ λr (1 λ), Φ sophisticated λr C, λφ λr C, Φ (1 λ)ρ Table IV Nash Equilibria. Dependent on the parameters of the model, there are four possible Nash equilibria in pure strategies, and one Nash equilibrium in mixed strategies with (α,β ) as in Equations (13) and (14) being the probabilities for being a trusting investor and an honest CRA, respectively. The Nash equilibria in pure strategies are labeled with the capital initials of the respective strategies, i.e., (T)rusting, (S)ophisticated, (H)onest, and (I)nflating. In brackets, I indicate the weakly dominated second Nash equilibrium, if applicable. The numbers indicate the sections in which the respective cases are treated for the evolutionary model. Φ/C C =0 0 <C<1 λ C 1 λ α Φ=0 S/H (T/H), 3.5 T/H, 3.4 T/H (T/I), 3.4 α =1 0 < Φ <ρ S/H, 3.5 (α,β ), 3.1 T/I, <α < 1 Φ ρ S/I (S/H), 3.2 S/I, 3.2 T/I (S/I), 3.3 α 0 β β =1 0 <β < 1 β 0 probabilities (α,β ) for being a trusting investor and an honest CRA, respectively. This means that each player randomizes such that the other player is indifferent between the available strategies. More precisely, the investor chooses α such that ΔΠ CRA =Π CRA H Similarly, the CRA chooses β such that ΔΠ Inv =Π Inv T Π CRA I =0 α = α := 1 Φ ρ. (13) Π Inv S =0 β = β := 1 C 1 λ. (14) Moreover, as Table IV shows, there are four possible Nash equilibria in pure strategies: 11 First, if Φ ρ and C < 1 λ, the equilibrium is sophisticated/inflating. For the CRA, the fees charged outweigh the possible reputation costs, and for the investor, the monitoring cost is low relative to the share of bad investments, so monitoring pays off. If in this first case, Φ = ρ and C = 0, then there is a second Nash equilibrium sophisticated/honest, which is weakly dominated by sophisticated/inflating. This means that it is equally good for the CRA to be honest or inflating given that the investor is sophisticated. However, if the investor chooses the off-equilibrium strategy to be trusting, then the CRA is better off being inflating. Second, if C 1 λ and Φ > 0, the equilibrium is trusting/inflating. In this case, the investors do better not to monitor the investment quality, because there are relatively 11 I thank Henri Pagès for suggesting a superior exposition of the Nash equilibria.

11 Credit Rating Dynamics and Competition 11 few bad investments on the market. Even with an inflating CRA and the investor buying all investments, the average return on the investment exceeds the monitoring costs. If in this second case, Φ = ρ and C =1 λ, then there is a second Nash equilibrium sophisticated/inflating, which is weakly dominated by trusting/inflating. This means that it is equally good for the investor to be trusting or sophisticated given that the CRA is inflating. However, if the CRA chooses the off-equilibrium strategy to be honest, then the investor is better off being trusting. Third, if Φ = 0 and C>0, the equilibrium is trusting/honest. Here, there is no fee differential for good and bad ratings, and thus no incentive for the CRA to inflate ratings. The investor, on the other hand, can encounter such a CRA in trusting mood and does not have to spend costs on monitoring quality. If in addition to Φ = 0, it even holds that C>1 λ, then there is a second Nash equilibrium trusting/inflating, which is weakly dominated by trusting/honest. This means that it is equally good for the CRA to be honest or inflating given that the investor is trusting. However, if the investor chooses the off-equilibrium strategy to be sophisticated, then the CRA is better off being honest. Fourth, if C = 0 and Φ < ρ, the equilibrium is sophisticated/honest. If the investor does not have to spend monitoring costs, it is the best choice to be sophisticated and thus deter inflating CRA behavior. If in addition to C = 0 it even holds that Φ = 0, then there is a second Nash equilibrium trusting/honest, which is weakly dominated by sophisticated/honest. This means that it is equally good for the investor to be trusting or sophisticated given that the CRA is honest. However, if the CRA chooses the offequilibrium strategy to be inflating, then the investor is better off being sophisticated. As I will show in the following, the outcomes of the evolutionary dynamics are related to the outcomes of the two-player game presented in the current section. For the interior case with 0 <C<1 λ and 0 < Φ <ρ, I will show that the outcome is a cyclic dynamic behavior around a fixed point at (α,β ). For the other cases, I show that the Nash equilibrium outcomes of the two-player game are also reached similarly as equilibria in the dynamic setting. The evolutionary game perspective allows to make predictions about the real-world situation in which not only a single investor and a single CRA, but populations of investors and CRAs interact over time. Although mathematically similar to the Nash equilibrium outcomes of the two-player game, only evolutionary dynamics allow the interpretation of interacting populations, rather than pure and mixed strategies of two single players. A two-player game does not describe the market well, and especially the interpretation of (α,β ) as a fixed point of cyclic dynamics is completely different from a Nash equilibrium in mixed strategies. Also, the CRA payoff definitions are based on ratings shopping and thus only have a meaningful interpretation on a market with several CRAs. Moreover, I will discuss in Section 4. how the outcomes change if the model s parameters are dependent on the number of CRAs in the market. 2.5 EVOLUTIONARY GAME From now on, I assume that there are interactions between a large number of individual investors and CRAs. For each interaction, there is a random draw of one investor and one CRA with built-in types out of their respective populations. The population shares change over time as a result of the payoffs achieved in the interactions. For such a situation, the corresponding replicator dynamics can be derived as in Taylor and Jonker (1978) and

12 12 Stefan Hirth Taylor (1979) to be 12 α t = α(πinv T Π Inv ) and β t = β(πcra H Π CRA ). (15) This means that the growth rate α /α of the trusting investors population share equals t the difference between the trusting investors current payoff and the current average payoff in the investor population. If trusting investors perform better than average, their share is growing. Formerly sophisticated investors and new market entrants will adopt the successful behavior of being trusting. If trusting investors perform worse than average, their share is shrinking. The opposite holds for the share (1 α) of sophisticated investors, respectively, and an analogous mechanism is at work in the CRA population. Note that there is some stickiness in the behavior. The intuition behind is that it takes some time for the knowledge about the success of the respective strategy to reach all the market participants. The dynamics can be transformed into and β t α t = α(1 α)(πinv T Π Inv S ) with ΔΠ Inv = C (1 β)(1 λ) (16) }{{} ΔΠ Inv = β(1 β)(πcra H Π CRA I ), with ΔΠ CRA =(1 λ)((1 α)ρ Φ). (17) }{{} ΔΠ CRA The transformation allows the following interpretation: When trusting investors perform relatively better than sophisticated investors, then the trusting investors share α in the investor population is increasing. Similarly, when honest CRAs perform relatively better than inflating CRAs, then the honest CRAs share β in the CRA population is increasing. As there are only two strategies in each population, better than average is equivalent to better than the other strategy. My next step is to derive stationary regions. These refer to states in which there is no movement in either α or β direction, or neither. There is no movement in α direction, if α = 0 in Equation (16). This is the case if either α =0orα = 1 (on the margins of the t population space), or ΔΠ Inv =0 β = β := 1 C 1 λ. (18) Similarly, there is no movement in β direction, if β = 0 in Equation (17). This is the case t if either β =0orβ = 1 (again, on the margins of the population space), or ΔΠ CRA =0 α = α := 1 Φ ρ. (19) Note that the solutions for α and β are the same as those derived in Equations (13) and (14) as probabilities for the Nash equilibrium in mixed strategies. For states above or below the fixed lines, I observe from Equation (16) that ΔΠ Inv is increasing in β. Therefore, I have for β>β (<) a movement α 0( 0). Similarly, I observe from Equation (17) t that ΔΠ CRA is decreasing in α. Likewise, I have for α>α (<) a movement β 0( 0). t If there is no movement in either direction, then the corresponding state is a fixed point in the dynamics. From the required condition α = β = 0, I derive the corners of t t 12 See Appendix B for more details on the derivation.

13 Credit Rating Dynamics and Competition 13 the population space as fixed points, as well as the interior fixed point (α,β ). For the existence of an interior fixed point, I need α,β (0, 1). From Equations (18) and (19), this means that 0 < 1 C < 1and0< 1 Φ < 1, or reformulated, 1 λ ρ and 0 <C<1 λ (20) 0 < Φ <ρ. (21) If the interior fixed point exists, then it is a center, as shown in the Appendix, Section C. This means that the orbits will spiral periodically around the fixed point, keeping their radius and speed constant. 2.6 INVESTOR WELFARE As investor welfare, I define the current average payoff Π Inv in the population of investors given by Equation (6), which can be rewritten as Π Inv = λr [(1 α)c + α(1 β)(1 λ) ]. (22) }{{}}{{} An intuitive interpretation is that welfare is created by investment into good projects, which generates a value of λr in expectation. Deductions occur for two reasons. First, there are the monitoring costs (1 α)c that sophisticated investors spend to avoid buying bad investments. Second, there are the losses α(1 β)(1 λ) occuring upon the default of bad investments bought by trusting investors and rated as good by inflating CRAs. The usual first-order condition for a maximum with respect to the share of trusting investors α is Inv Π = C (1 β)(1 λ) =0, (23) α whereas = 0 shows that further analysis is needed. Note that =ΔΠ Inv from Equation (16), therefore solving Equation (23) for β yieldsthesamesolutionβ =1 C 1 λ as in Equation (18). Given that there is a share β of honest CRAs on the market, the corresponding investor welfare is Π Inv (β )=λr C. Furthermore, Equation (23) says that for β higher (lower) than β, the investor welfare is increasing (decreasing) in α, with the boundaries α =1 (= 0) being the solutions maximizing investor welfare at values of λr (1 β)(1 λ) and λr C, respectively. Comparing to the dynamics from Equation (16), note that whenever β>β induces an increase of trusting investors in the market over time, then investor welfare also increases with the share of trusting investors. However, it is important to note that the behavior of investors and CRAs is mutually dependent. While the investor behavior is changing over time, the CRA behavior is as well. Particularly, in the base case with an interior fixed point and cyclic dynamics, the boundary solutions α = 1 (= 0) that maximize investor welfare will generally not be reached. Therefore I will look at the joint dynamic evolution of population shares in the following section. 2 Π Inv α 2 Π Inv α

14 14 Stefan Hirth Table V Base Case Parameter Values. The table states the numerical values chosen for each of the parameters in the base case of the model. share of good investments λ =0.5 net payoff upon investment R = 0.5 verification cost C = 0.2 fee charged by CRA Φ=1 reputation cost ρ = Results for an Arbitrary Number of CRAs The equilibrium resulting from the interaction of an arbitrary number of CRAs and investors depends on the parametrization of the model. More precisely, it depends on whether an interior fixed point (α,β ) exists, which is the case whenever Equations (20) and (21) are satisfied. If (α,β ) lies outside the population space, it still makes a difference on which side it lies. This results in several interesting cases, which will be analyzed in the following sections. 3.1 BASE CASE: INTERIOR FIXED POINT AND CYCLES As base case I define the situation with one interior fixed point at (α,β ), i.e., α,β (0, 1). To satisfy Equations (20) and (21) derived in the previous section, I choose the share of good investments as λ = 0.5, the verification cost borne by sophisticated investors as C =0.2, the fee charged by the CRA to the issuer for a good rating as Φ = 1, and the CRA s reputation cost if caught lying as ρ = 1.4. The net payoff upon investment is chosen as R =0.5. Note that the latter affects neither the location of (α,β ) nor the dynamics in Equations (16) and (17). These are driven only by the differences in payoffs between strategies, and the investor payoffs can all be shifted by λr to eliminate R. The parameter values are summarized in Table V. According to Equations (18) and (19), the resulting interior fixed point is located at (α,β )=(0.29, 0.6). Note that the qualitative properties of the model depend only on whether α and β lie in the interior of the population space. Therefore the choice of parameters serves purely illustrative purposes. The resulting dynamics are visualized in Figure 1. The arrows indicate the vectors ), i.e., the direction and speed of development of the shares in each population for given current shares of trusting investors (α) and honest CRAs (β). Also shown are the fixed lines on the margins of the population space, and the two interior fixed lines. The latter separate four different regimes. Dependent on the current investor sophistication level, either the honest or inflating CRAs are more successful. For example, start in the middle of the population space, at (0.5,0.5), which, for the given parametrization, lies in the lower right quadrant of the population space. Then the share of trusting investors (α) is higher than on the fixed line (α ). As a consequence, the inflating CRAs are the more successful ones, and their share is growing. Also, the current state has a level of honest CRAs (β) thatisbelowthefixedline(β ). Therefore the trusting investors often happen to put their money in bad investments. The sophisticated investors are better off in such a situation and can improve their market share. As the arrows indicate, the combined effect is that the market moves towards less honest (and more inflating) CRAs and less trusting (and more sophisticated) investors. ( α, β t t

15 Credit Rating Dynamics and Competition 15 Β 1 1 Α Fig. 1. Base case: interior fixed point and cycles. The figure shows the dynamic evolution of the population shares, with α representing the share of trusting (rather than sophisticated) investors and β representing the share of honest (rather than inflating) CRAs. It is displayed as a vector field for α,β (0, 1). The fixed lines α and β are indicated by vertical and horizontal lines, respectively. Parameter values correspond to the base case parametrization given in Table V. Once the fixed line at α is crossed, the lower left quadrant of the population space is entered. Now the share of trusting investors is still shrinking, because of the high market share of inflating CRAs. However, now there are enough sophisticated investors in the market to make reputation costs more important for the CRAs than rating fees. Therefore the honest CRAs make more profit now, and they consequently improve their market share. When crossing the fixed line at β, yet another regime is reached in the upper left quadrant of the population space. Now the share of honest CRAs is high enough that it does not pay off anymore for the individual investor to invest in the monitoring of the CRAs. Therefore the trusting investors now perform better than the sophisticated ones and gain in market share. Still, there are enough sophisticated investors in the market to make the honest CRAs better off than the inflating ones, and thus the honest CRAs market share is further growing. In the next regime transition, the fixed line at α is crossed again, and the upper right quadrant of the population space is entered. Here, the trusting investors still become more numerous. Due to the low share of the inflating CRAs, it is still safe to buy all investments rated as good, rather than investing in monitoring. However, the trusting investors have already become such a big group that again it is beneficial for the CRAs to inflate ratings, as they can do so with little risk of being caught. Therefore the inflating CRAs gain market

16 16 Stefan Hirth share on expense of the honest ones. Finally, the last transition over the fixed line at β leads again into the lower right quadrant of the population space, where I started the investigation. As previously derived, the fixed point (α,β ) is a center in the dynamics. This means that the clockwise cycles of movement in the population that are displayed in Figure 1 will spiral periodically around the fixed point, rather than move towards or away from it. These cycles of movement are consistent with evidence of both CRAs and investors behavior varying over the business cycle. Related are the following two theoretical predictions: Bar-Isaac and Shapiro (2012) find that ratings accuracy is counter-cyclical. BFS predict that ratings inflation is more likely in boom times when investors have lower incentives to perform due diligence, as the ex ante quality of investments is then higher. In the language of my model, this corresponds to a higher share of inflating CRAs for a lower share of sophisticated investors in the market. The effect of the ex-ante investment quality will be analyzed in Section 3.3. The conclusion for the base case depends on the current regime of the market. It pays off temporarily for CRAs to be honest, but only if there are enough sophisticated investors in the market, who make reputation loss a real threat. Otherwise, ratings inflation is the best strategy for the CRAs. My result for the base case is consistent with other theories, e.g., Skreta and Veldkamp (2009) and BFS, who suggest that ratings inflation is most severe for complex investment products, i.e., products exceeding the investor sophistication level, and for more trusting investors. However, in contrast to the existing theories I emphasize that even for the base case of my model, strategic honesty can pay off at least temporarily. Investor welfare in the base case is varying over time according to Equation (22), driven by the changing population shares. For β = β as in the center of the dynamics, the corresponding investor welfare is Π Inv (β )=λr C. Apart from the base case, I highlight several other cases in the following. These occur for parameter constellations in which one or both of the coordinates (α,β ) lie outside of the population space. Interestingly, very different outcomes will be reached for these cases, including stable equilibria in which the competitive market for credit ratings will be served exclusively by honest CRAs. The analysis of these cases will also allow policy recommendations on how to induce honest rating behavior. 3.2 SOPHISTICATED INVESTORS AND INFLATING CRAS I begin with describing the equilibrium that occurs if Φ >ρ. This means that the inflating CRA, if caught, faces reputation costs below the fees earned. Then it follows from Equation (19) that α < 0. The dynamics are visualized in Figure 2(a) given that 0 <C<1 λ (the resulting equilibrium is the same for C =0). The resulting equilibrium is that the honest CRAs die out, and trusting investors die out as well. The corresponding investor welfare according to Equation (22) is V SH = λr C. From an investor point of view, it pays off to collect information oneself instead of listening to the CRAs. The sophisticated investors can thus avoid to buy bad investments. Still, the issuers pay fees to the rating agencies. As these fees are higher than the reputation costs, it is still worth for the rating agencies to produce useless information and get paid for it. A possible critique is that even if there are no trusting investors on the market (α =0), the CRAs still receive fees from the issuers. Instead, the issuers might want to make sure